Find the slope of the line given by $\theta =\frac{3\pi }{4}$ when expressed in Cartesian coordinates.

Find the slopes of the lines tangent to the polar curve $r\left(\theta \right)=4+8\cos \theta$ at the origin.

Determine all intersection points (with $\theta \in [0,2\pi )$) of the polar curves $r=1$ and $r=2\sin \theta$.

Convert the circle $x^2+(y-3{)}^2=9$ into polar coordinates.

Convert the polar equation $r^2+r(10\cos \theta -6\sin \theta )=0$ to Cartesian form and identify the curve.